In [4], a subgroup theorem for HNN groups was established. The theorem was proved by embedding the given HNN group in a free product with amalgamated subgroup and then applying the subgroup theorem of [3]. In this paper we obtain a sharper form of the subgroup theorem of [4] by applying the Reidemeister-Schreier method directly, using an appropriate Schreier system of coset representatives. Specifically, we prove (in Theorem 1) that if H is a subgroup of the HNN group

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A Steiner triple system is a pair (Q, t) where Q is a set and t a collection of three element subsets of Q such that each pair of elements of Q belong to exactly one triple of t. The number |Q| is called the order of the Steiner triple system (Q, t). It is well-known that there is a Steiner triple system of order n if and only if n ≡ 1 or 3 (mod 6). Therefore in saying that a certain property concerning Steiner triple systems is true for all n it is understood that n ≡ 1 or 3 (mod 6). Two Steiner triple systems (Q, t1) and (Q, t2) are said to be disjoint provided that t1 ∩ t2 = Ø. Recently, Jean Doyen has shown the existence of a pair of disjoint Steiner triple systems of order n for every n ≧ 7 [1].

In 1963, P. L. Butzer [4, p. 180] asked whether it was possible to prove Jackson's theorem by means of an operator which is "almost" interpolatory in the sense that it is based on the values of the approximee at a finite number of nodes. In answer to this question, G. Freud introduced [4] a sequence of operators which led to an independent proof of Jackson's theorem. Strictly speaking these operators are not interpolatory but they are "almost" interpolatory in the above sense.

Recently R. G. Douglas showed [4] that if V is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinite dimensional Hilbert space ), then V — K is unitarily equivalent to V ⊕ U (acting on ⊕ ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary).

With the aid of a theorem about the Julia points of a function meromorphic in the unit disk, this paper strengthens a theorem of K. Meier. As a consequence a stronger form of Plessner's Theorem is seen to hold which contains a theorem of E. F. Collingwood. An additional consequence is a stronger form of Meier's analogue to Plessner's Theorem.